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arXiv · 2606.22323

Bloch's conjecture for equivalences between twisted abelian surfaces and applications

Zaiyuan Chen·Zhiyuan Li·Ruxuan Zhang
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Abstract

The Beauville--Voisin conjecture predicts a canonical descending filtration on the Chow group of zero-cycles of a hyperk\"ahler variety, opposite to the conjectural Bloch--Beilinson filtration. A basic test for such filtrations is a Bloch-type principle: the action on zero-cycles should be governed by the action on the holomorphic symplectic form. While this principle has been verified in several cases of hyperk\"ahler varieties of $\mathrm{K3}^{[n]}$-type, the $\mathrm{Kum}_n$-type case remains much less understood. In this paper, we study this problem through twisted abelian surfaces and their associated $\mathrm{Kum}_n$-type varieties. We first construct a natural action of autoequivalences of twisted abelian surfaces on the Albanese kernel and prove Bloch's conjecture for all (anti-)symplectic autoequivalences. As an application, we prove the corresponding Bloch conjecture for symplectic birational automorphisms of twisted modular $\mathrm{Kum}_n$-type varieties; in particular, this applies to those admitting a birational Lagrangian fibration. Finally, we introduce and study a Shen--Yin--Zhao type filtration on twisted modular varieties and compare it with Voisin's filtration in the sixfold case. We also establish the anti-symplectic Bloch conjecture for twisted modular $\mathrm{Kum}_3$-type varieties.

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Zaiyuan Chen, Zhiyuan Li, Ruxuan Zhang. 2026-06-21. Bloch's conjecture for equivalences between twisted abelian surfaces and applications. https://arxiv.org/abs/2606.22323

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